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Gröbner bases over fields with valuation and tropical curves by coordinate projections
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Chan, Andrew John (2013) Gröbner bases over fields with valuation and tropical curves by coordinate projections. PhD thesis, University of Warwick.
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WRAP_THESIS_Chan_2013.pdf - Submitted Version Download (981Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b2698876~S1
Abstract
In the emerging field of tropical geometry, algebraic varieties are replaced by
polyhedral objects called tropical varieties. The algebraic and tropical variety share
many invariants, but due to its polyhedral structure the tropical variety is often
easier to work with. In this thesis, we look at two problems related to constructing
tropical varieties.
In the first, we extend the theory of Grobner bases to the case where we
are looking over a field with a valuation. The motivation is that we can use these
Grobner bases in order to compute tropical varieties over fields with valuations. We
discuss some complexity and implementation issues and present a family of ideals
whose Grobner basis with respect to the p-adic valuation is small, but all of whose
standard Grobner bases are large.
In the second, we investigate finding tropical curves over fields with the
trivial valuation from their two-dimensional coordinate projections. A tropical curve
has the support of a one-dimensional fan, and we use its coordinate projections to
reconstruct the rays of this fan. We discuss some implementation issues and we see
examples of tropical curves which can be computed using our projection techniques
which cannot be computed with existing techniques.
Item Type: | Thesis (PhD) |
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Subjects: | Q Science > QA Mathematics |
Library of Congress Subject Headings (LCSH): | Gröbner bases, Tropical geometry |
Official Date: | August 2013 |
Institution: | University of Warwick |
Theses Department: | Mathematics Institute |
Thesis Type: | PhD |
Publication Status: | Unpublished |
Supervisor(s)/Advisor: | Maclagan, Diane, 1974- |
Extent: | vi, 108 leaves : illustrations |
Language: | eng |
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